# anova.mlm

##### Comparisons between Multivariate Linear Models

Compute a (generalized) analysis of variance table for one or more multivariate linear models.

- Keywords
- multivariate, models, regression

##### Usage

```
# S3 method for mlm
anova(object, …,
test = c("Pillai", "Wilks", "Hotelling-Lawley", "Roy",
"Spherical"),
Sigma = diag(nrow = p), T = Thin.row(proj(M) - proj(X)),
M = diag(nrow = p), X = ~0,
idata = data.frame(index = seq_len(p)), tol = 1e-7)
```

##### Arguments

- object
- an object of class
`"mlm"`

. - …
- further objects of class
`"mlm"`

. - test
- choice of test statistic (see below). Can be abbreviated.
- Sigma
- (only relevant if
`test == "Spherical"`

). Covariance matrix assumed proportional to`Sigma`

. - T
- transformation matrix. By default computed from
`M`

and`X`

. - M
- formula or matrix describing the outer projection (see below).
- X
- formula or matrix describing the inner projection (see below).
- idata
- data frame describing intra-block design.
- tol
- tolerance to be used in deciding if the residuals are
rank-deficient: see
`qr`

.

##### Details

The `anova.mlm`

method uses either a multivariate test statistic for
the summary table, or a test based on sphericity assumptions (i.e.
that the covariance is proportional to a given matrix). For the multivariate test, Wilks' statistic is most popular in the
literature, but the default Pillai--Bartlett statistic is
recommended by Hand and Taylor (1987). See
`summary.manova`

for further details. For the `"Spherical"`

test, proportionality is usually with the
identity matrix but a different matrix can be specified using `Sigma`

).
Corrections for asphericity known as the Greenhouse--Geisser,
respectively Huynh--Feldt, epsilons are given and adjusted \(F\) tests are
performed. It is common to transform the observations prior to testing. This
typically involves
transformation to intra-block differences, but more complicated
within-block designs can be encountered,
making more elaborate transformations necessary. A
transformation matrix `T`

can be given directly or specified as
the difference between two projections onto the spaces spanned by
`M`

and `X`

, which in turn can be given as matrices or as
model formulas with respect to `idata`

(the tests will be
invariant to parametrization of the quotient space `M/X`

). As with `anova.lm`

, all test statistics use the SSD matrix from
the largest model considered as the (generalized) denominator. Contrary to other `anova`

methods, the intercept is not excluded
from the display in the single-model case. When contrast
transformations are involved, it often makes good sense to test for a
zero intercept.

##### Value

An object of class `"anova"`

inheriting from class `"data.frame"`

##### Note

The Huynh--Feldt epsilon differs from that calculated by SAS (as of v. 8.2) except when the DF is equal to the number of observations minus one. This is believed to be a bug in SAS, not in R.

##### References

Hand, D. J. and Taylor, C. C. (1987)
*Multivariate Analysis of Variance and Repeated Measures.*
Chapman and Hall.

##### See Also

##### Examples

`library(stats)`

```
require(graphics)
utils::example(SSD) # Brings in the mlmfit and reacttime objects
mlmfit0 <- update(mlmfit, ~0)
### Traditional tests of intrasubj. contrasts
## Using MANOVA techniques on contrasts:
anova(mlmfit, mlmfit0, X = ~1)
## Assuming sphericity
anova(mlmfit, mlmfit0, X = ~1, test = "Spherical")
### tests using intra-subject 3x2 design
idata <- data.frame(deg = gl(3, 1, 6, labels = c(0, 4, 8)),
noise = gl(2, 3, 6, labels = c("A", "P")))
anova(mlmfit, mlmfit0, X = ~ deg + noise,
idata = idata, test = "Spherical")
anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ noise,
idata = idata, test = "Spherical" )
anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ deg,
idata = idata, test = "Spherical" )
f <- factor(rep(1:2, 5)) # bogus, just for illustration
mlmfit2 <- update(mlmfit, ~f)
anova(mlmfit2, mlmfit, mlmfit0, X = ~1, test = "Spherical")
anova(mlmfit2, X = ~1, test = "Spherical")
# one-model form, eqiv. to previous
### There seems to be a strong interaction in these data
plot(colMeans(reacttime))
```

*Documentation reproduced from package stats, version 3.3.3, License: Part of R 3.3.3*